2017/2018 SEMESTER 1 COMMON TEST

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1 07/08 SEMESTER COMMON TEST Course : Diplom i Electroics, Computer & Commuictios Egieerig Diplom i Electroic Systems Diplom i Telemtics & Medi Techology Diplom i Electricl Egieerig with Eco-Desig Diplom i Mechtroics Egieerig Diplom i Digitl & Precisio Egieerig Diplom i Aerouticl & Aerospce Techology Diplom i Biomedicl Egieerig Diplom i Notechology & Mterils Sciece Module : EG008 / EG68 / EG96 EGB07 / EGD07 / EGF07 EGH07 / EGJ07 Egieerig Mthemtics B Egieerig Mthemtics B Mthemtics B Jue 07 Time Allowed:.5 hrs INSTRUCTIONS TO CANDIDATES This test pper cosists of Nietee (9) pges icludig this pge. Aswer ALL questios. Suggested Solutios All solutios must be writte clerly i the spces provided. DO NOT write i pecil. 4 Mthemticl formul tbles re provided o pges 5 to 6. Admissio Number: Module Group: Questio Number Mrks Questio Number Mrks /0 4 /0 /0 5 /0 /0 6 /0 Totl : /00

2 EGJ07/ Pge Questio A survey ws coducted o twelve differet people o their mothly slries (i dollrs) d the result of the survey is show below () For the smple dt give bove, fid the followig: (i) Me. ( mrk ) (ii) Medi. ( mrks ) (iii) Stdrd devitio. ( mrk ) (iv) Iterqurtile rge. ( mrks ) (b) Epli whether the smple me or medi is better mesure of cetrl tedecy for this set of dt. ( mrks ) Aswer Solutios () (i) Usig clcultor, = (ii) Arrge umbers i scedig order 750, 800, 900, 950, 050, 00, 50, 50, 00, 400, 450, 600 Medi, Q = = 5 (iii) Usig clcultor, s = (iv) Q = = 95, Q = = 50 Iterqurtile rge = = 45 (b) Medi is better mesure s it is ot ffected by the outlyig vlue, 600.

3 EGJ07/ Pge Questio The sles mger of lrge retiler of electricl pplices is moitorig the effects of televisio dvertisig cmpig o the sles of their pplices. The televisio time, i miutes, which hve bee purchsed for the cmpig d the correspodig sles y, i hudreds, of pplices for the lst seve weeks re recorded. The sctter digrm d the lier regressio summry output is give below. 0 Sles (y) vs Advertisig time ()

4 EGJ07/ Pge 4 () Stte the correltio coefficiet d describe the reltioship betwee the umber of sles d the dvertisig time. ( 4 mrks ) (b) Stte the equtio of the best fit lie i the form of y = + b d describe i cotet wht do the vlues of d b represet. ( 8 mrks ) (c) Usig the equtio of the best fit lie i (b), estimte the umber of sles whe the dvertisemet is 6 miutes log. Commet o the relibility of the estimte. ( 5 mrks ) (d) Suppose there is ew dvertisemet tht promises to icrese the sles by ectly 00 compred to the origil dvertisemet. Write dow the equtio of the ew best fit lie d clculte the estimted umber of sles whe the dvertisemet is miutes log. ( mrks ) Solutios () r = The reltioship is positive, very strog, lier. (b) y = The umber of sles icreses pproimtely by 75.5 whe the dvertisemet time icreses by miute. The umber of sles is pproimtely 5 whe there is o dvertisemet. (c) Whe 6 =, umber of sles [ ] = (6) The estimtio is relible s r close to d = 6 is withi the dt rge.

5 EGJ07/ Pge 5 THIS IS AN ANSWER SHEET FOR QUESTION (d) y = Whe =, umber of sles [ ] = ()

6 EGJ07/ Pge 6 Questio Ech lphbet i the word STATISTICS is writte dow o seprte crd. Fid the umber of wys to () rrge ll the crds without y restrictio, ( mrks ) (b) rrge ll the crds such tht the sme lphbets re together, ( mrks ) (c) form word usig three of the crds. ( 6 mrks ) Solutios () 0! 50400!!! = (b) 5! = 0 (c) Cse : All letters differet C! = 60 5 Cse : Ectly pir of ideticl letters! C C = 6! 4 Cse : ideticl letters C = Totl umber of wys = = 89

7 EGJ07/ Pge 7 Questio 4 () (i) Give tht X ~ B (5,0.), clculte PX ( ). ( mrks ) (ii) Give tht Y ~ Po (), clculte PY ( ). ( mrks ) (b) Suppose vehicles rrive t siglised rod itersectio t verge rte of 60 per hour d the cycle of the trffic light is set t 0 secods. Fid the probbility tht the umber of vehicles rrivig i 0 secods itervl will be (i) ectly, ( mrks ) (ii) less th? ( 4 mrks ) (iii) After the lights chge to gree, there is time to cler oly vehicles before the sigl chges to red gi. Show tht the probbility of the witig vehicles re ot clered i oe cycle is ( mrks ) A ispectio is coducted o 0 cosecutive cycles of the trffic light. (iv) Usig the swer i (iii), fid the probbility tht there re 8 cycles with witig crs ot clered. ( mrks ) (v) Fid the me d vrice of the umber of cycles with witig crs ot clered. ( mrks ) Solutios ( = 0) + ( = ) = () (i) PX PX C( ) ( ) C( ) ( ) 0 (ii) 0 PY PY e ( ) = ( = 0) = !

8 EGJ07/ Pge 8 THIS IS AN ANSWER SHEET FOR QUESTION 4 (b) (i) Let X be the umber of vehicles rrivig i 0 secods. PX e ( = ) = 0.4! X ~ Po ( ) (ii) PX ( < ) = PX ( = 0) + PX ( = ) + PX ( = ) 0 e = + + 0!!! 0.4 (iii) PX ( > ) = PX ( ) = 0.4 = (iv) Let Y be the umber of cycles with witig crs ot clered. Y ~ B ( 0, ) ( ) ( ) 0 8 ( = 8) = C PY (v) EY ( ) = Vr( Y ) = 0(0.5768)( ) 4.88

9 EGJ07/ Pge 9 Questio 5 () (i) Give tht X ~ N (0, 4), clculte PX< ( ). ( mrks ) (ii) Give tht Y ~ N ( µ,) d PY< ( ) = , fid µ. ( 4 mrks ) (b) The mss of coffee i rdomly chose jr sold my be tke to hve orml distributio with me 0 grms d stdrd devitio of.5 grms. (i) (ii) Fid the probbility tht rdomly chose jr will coti t betwee 00 d 05 grms of coffee. ( 5 mrks ) Fid the mss m (i grms) such tht oly % of jrs coti more th m grms of coffee. ( 4 mrks ) Solutios (iii) Fid the probbility tht eight jrs, o verge, will coti less th 0 grms of coffee. ( 4 mrks ) () (i) (ii) 0 PX ( < ) = P Z< = PZ ( <.5) = 0.9 µ PY ( < ) = P Z< = µ = 0.5 µ = 0.5.6

10 EGJ07/ Pge 0 THIS IS AN ANSWER SHEET FOR QUESTION 5 (b) (i) Let W be the mss of coffee i jr. W ~ N ( 0,.5 ) ( ) ( ) P(00 < W< 05) = PW< 05 PW = P Z < P Z.5.5 ( 0.8) P( Z.) = P Z < = = 0.67 (ii) PW ( m) = 0.97 m 0 P Z = m 0 =.88 m 08.5 (iii).5 W N ~ 0, ( 0) PW< = P Z<.5 / 8 (.) = P Z < = 0.9

11 EGJ07/ Pge Questio 6 A coi d three-fced die with umbers, d re throw simulteously. Ech umber o the die hs equl chce of beig observed. The rdom vrible X is defied s follows: If the coi shows hed, the X is the score o the die. If the coi shows til, the X is twice the score o the die. () Complete the tree digrm below. ( mrks ) (b) Fid the probbility tht X = give tht the coi shows hed. ( mrks ) (c) Complete the probbility distributio tble for X below. ( 6 mrks ) (d) Fid the probbility tht X is prime umber or more th 4. ( mrks ) (e) Clculte EX ( ) d Vr( X ). ( 6 mrks ) Aswer () H T (b) (c) X = k PX ( = k)

12 EGJ07/ Pge Solutios () THIS IS AN ANSWER SHEET FOR QUESTION 6 / / 0.5 / 0.5 / / / 4 6 (b) PX ( = H) = (c) X = k 4 6 PX ( = k) PX ( prime X> 4) = P Xprime + PX ( > 4) PX ( prime X> 4) (d) ( ) = + 0 = 6 (e) EX ( ) = = EX ( ) = = ( ) [ ] Vr( X ) = E X E( X ) = 8

13 EGJ07/ Pge Formul Tbles Rules o Differetitio/Itegrtio Product Rule: d du dv ( uv) = v + u, u, v re fuctios of Quotiet Rule: du dv v u d u = v v d d du Chi Rule: [ f ( u) ] = [ f ( u) ] du Itegrtio by Prts: u dv = uv vdu fg gf Specil Cse: fg =, m m where f = f, g = mg Stdrd Derivtives d ( ) = d d [ cf ( ) ] = c [ f ( ) ] c = costt d d ( si ) = cos, ( cos ) = si d ( t ) = sec, d ( cot ) = csc d d ( csc ) = csc cot, ( sec ) = sec t d ( e ) = e, d ( l ) = d ( cos ) =, d ( ) si = d ( t ) = + Stdrd Itegrls (The costt of itegrtio is omitted) + = ( ) + [ ] + [ f( ) ] f '( ) f ( ) = + + c ( ) = l '( ) f = l f( ) f ( ) si = cos si + cos = si cos ( + b) = cos( b) ( + b) = si( + b) = t sec + b = t + b sec t = sec sec ( ) ( ) csc = cot sec = + l sec t csc cot = csc sec l sec t ( + b) = ( + b) + ( + b) csc = l csc cot = l csc + cot csc( + b) = l csc( + b) cot( + b) or = l csc( + b) + cot( + b) e = e + b + b e = e = si = si = t + t = + + = l

14 EGJ07/ Pge 4 Arithmetic Progressio The th term, = + ( ) u d S = + d Sum of first terms, ( ) Geometric Progressio The th term, u = r Sum of first terms, = [ first term + lst term ] ( r ) S =, r 0 r Sum to ifiity =, < r < r Useful Trigoometric Fcts ( ) ( ) si π = si π = 0 ( π ) = ( π ) = ( ) cos cos Some Useful Trigoometric Idetities si θ+ cos θ= + t θ= sec θ + cot θ= csc θ Summtio Properties C C i = = C = C i= i i= i ( i± yi) = y i i i ± = = i= i Biomil Distributio X ~ B, p ( ) ( ) ( ) ( ) = p ( ) where cos cos si = si θ = cos θ θ= θ θ P X = = C p p, = 0,,,..., E X Vr X = pq q = p Me d Vrice of Rdom Vrible The Me of discrete rdom vrible X, µ = k P X = k ( ) The Vrice of discrete rdom vrible X is σ = k PX ( = k) µ Mesures of Cetrl Tedecy Smple me: = Smple vrice: ( ) s = Lier Regressio Lie y = m + C, m is the slope/grdiet d C is the y- itercept Coutig! Permuttio: Pr = ( r)!! k! k!... k!! Combitio: Cr = r! ( r)! Probbility ( E) P( E) =, E is evet & S smple spce S ( ) ( ) PE ( ) P( A B) = P( A) + P( B) P( A B) P( A B) P E =, E is complemet evet of E P ( AB) m = or P( A B) = P( AB) P( B) ( ) P B A, B mutully eclusive P( A B) = 0 A, B idepedet P( A B) = P( A) P( B) Poisso Distributio X ~ P o ( ) = ( ) ( µ ) e µ P( X = ) =, = 0,,,,...! E X µ Vr X = µ µ

15 EGJ07/ Pge 5 Tble : Stdrd Norml Distributio 0 z z z

16 EGJ07/ Pge 6 z 0 z z

17 EGJ07/ Pge 7 Tble : t Distributio t t t t t t t t t t Level of cofidece, c Oe til, α d.f. Two tils, α END OF PAPER